The quickest way to get to Koh Phangan from Bangkok is going to be by air. You have two choices here, either fly to Surat Thani and take the boat from there or fly to Koh Samui and take the boat from there. The best method is to fly to Koh Samui as the boat from there to Phangan is only 90 minutes or so. Ferries from Surat Thani (Donsak pier) will take in the neighborhood of 2 hours.

If you plan on taking the quickest route then you can fly to Koh Samui with either Bangkok Airways or Thai Airways. I personally recommend Bangkok Airways as the prices are 60-70% less than Thai Airways. Off season flights are as low as THB3600. Once you land in Samui stop by the limosine counter to get your boat and bus tickets for Koh Phangan.

Those preferring to fly to Surat Thani can get there flying with Thai Airways or Air Asia, again the price for Thai Airways is 30-40% higher so unless you have some reason to fly with them I would choose Air Asia. Also take note that when booking air tickets during the high season (Nov-Mar) it is recommended that you book several weeks in advance.